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The function/' is 7c(/^)-integrable if and only if/' o n is /*-integrable, and in
that case we have If d(7t(uj) = \ (f o n) dp.


(13.8.1)   Let (/,) be an increasing sequence of integrable functions. For
sup/, to be integrable, it is necessary and sufficient that sup \fn d^< + 00,

»                                                                                                                     n     J

and if this condition is satisfied we have


= T \fn ^

+     (-/.) dp =

Since   f* fn d\i > — oo  for  all w, we  have   f * (sup/,) dp = sup \fndfji

J                                                         J    \ n     i           n   J

(13.5.7), This already shows that the condition is necessary. Conversely, if
the condition is satisfied, let /= sup/, . Then, given any e > 0, there exists n


such that the function/—/, (which is defined almost everywhere because/
and/, are finite almost everywhere (13.6.4)) satisfies

J*(/-/n) dp ^ J/dAi + J*(-

(13.5.6).   But there exists a function ue «pfR(X) such that \\fn — u\ dp ^ s
(13.7.2), hence by (13.5.6) we have

[V- "I dp £   r\f-fn\ dll -I-   f*^ - Il| d|X ^ 26,

and the result follows by (13.7.2).

There is of course a corresponding theorem for decreasing sequences of
integrable functions, obtained by replacing/, by ~/rt in (13.8.1).

(1 3.8.2)   Let (/,) be any sequence of integrable functions. In order thatf— sup/,


should be integrable, it is necessary and sufficient that there should exist a
function g^Q such that |   g dp < +00 andfn ^ g almost everywhere.

The condition is obviously necessary (take g =/+). Conversely, if it is
satisfied, let gn =   sup /fc. Then gn is integrable (13.7.4) and/=sup#w.tion on X' with values in R, we have